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In the geometry of curves, an **orthoptic** is the set of points for which two tangents of a given curve meet at a right angle.

Examples:

- The orthoptic of a parabola is its directrix (proof: see below),
- The orthoptic of an ellipse is the director circle (see below),
- The orthoptic of a hyperbola is the director circle (in case of
*a*≤*b*there are no orthogonal tangents, see below), - The orthoptic of an astroid is a quadrifolium with the polar equation (see below).

Generalizations:

- An
**isoptic**is the set of points for which two tangents of a given curve meet at a*fixed angle*(see below). - An
**isoptic**of*two*plane curves is the set of points for which two tangents meet at a*fixed angle*. - Thales' theorem on a chord PQ can be considered as the orthoptic of two circles which are degenerated to the two points P and Q.

Any parabola can be transformed by a rigid motion (angles are not changed) into a parabola with equation . The slope at a point of the parabola is . Replacing x gives the parametric representation of the parabola with the tangent slope as parameter: The tangent has the equation with the still unknown n, which can be determined by inserting the coordinates of the parabola point. One gets

If a tangent contains the point (*x*_{0}, *y*_{0}), off the parabola, then the equation
holds, which has two solutions *m*_{1} and *m*_{2} corresponding to the two tangents passing (*x*_{0}, *y*_{0}). The free term of a reduced quadratic equation is always the product of its solutions. Hence, if the tangents meet at (*x*_{0}, *y*_{0}) orthogonally, the following equations hold:
The last equation is equivalent to
which is the equation of the directrix.

Let be the ellipse of consideration.

- The tangents to the ellipse at the vertices and co-vertices intersect at the 4 points , which lie on the desired orthoptic curve (the circle ).
- The tangent at a point of the ellipse has the equation (see tangent to an ellipse). If the point is not a vertex this equation can be solved for y:

Using the abbreviations

(I) |

and the equation one gets: Hence

(II) |

and the equation of a non vertical tangent is
Solving relations **(I)** for and respecting **(II)** leads to the slope depending parametric representation of the ellipse:
(For another proof: see Ellipse § Parametric representation.)

If a tangent contains the point , off the ellipse, then the equation holds. Eliminating the square root leads to which has two solutions corresponding to the two tangents passing through . The constant term of a monic quadratic equation is always the product of its solutions. Hence, if the tangents meet at orthogonally, the following equations hold:

The last equation is equivalent to
From **(1)** and **(2)** one gets:

The intersection points of orthogonal tangents are points of the circle .

The ellipse case can be adopted nearly exactly to the hyperbola case. The only changes to be made are to replace with and to restrict m to |*m*| > *b*/*a*. Therefore:

The intersection points of orthogonal tangents are points of the circle , where *a* > *b*.

An astroid can be described by the parametric representation
From the condition
one recognizes the distance α in parameter space at which an orthogonal tangent to **ċ**(*t*) appears. It turns out that the distance is independent of parameter t, namely *α* = ± π/2. The equations of the (orthogonal) tangents at the points **c**(*t*) and **c**(*t* + π/2) are respectively:
Their common point has coordinates:
This is simultaneously a parametric representation of the orthoptic.

Elimination of the parameter t yields the implicit representation
Introducing the new parameter *φ* = *t* − 5π/4 one gets
(The proof uses the angle sum and difference identities.) Hence we get the polar representation
of the orthoptic. Hence:

The orthoptic of an astroid is a quadrifolium.

Below the isotopics for angles *α* ≠ 90° are listed. They are called α-isoptics. For the proofs see below.

- Parabola:

The α-isoptics of the parabola with equation *y* = *ax*^{2} are the branches of the hyperbola
The branches of the hyperbola provide the isoptics for the two angles α and 180° − *α* (see picture).

- Ellipse:

The α-isoptics of the ellipse with equation *x*^{2}/*a*^{2} + *y*^{2}/*b*^{2} = 1 are the two parts of the degree-4 curve
(see picture).

- Hyperbola:

The α-isoptics of the hyperbola with the equation *x*^{2}/*a*^{2} − *y*^{2}/*b*^{2} = 1 are the two parts of the degree-4 curve

- Parabola:

A parabola *y* = *ax*^{2} can be parametrized by the slope of its tangents *m* = 2*ax*:

The tangent with slope m has the equation

The point (*x*_{0}, *y*_{0}) is on the tangent if and only if

This means the slopes *m*_{1}, *m*_{2} of the two tangents containing (*x*_{0}, *y*_{0}) fulfil the quadratic equation

If the tangents meet at angle α or 180° − *α*, the equation

must be fulfilled. Solving the quadratic equation for m, and inserting *m*_{1}, *m*_{2} into the last equation, one gets

This is the equation of the hyperbola above. Its branches bear the two isoptics of the parabola for the two angles α and 180° − *α*.

- Ellipse:

In the case of an ellipse *x*^{2}/*a*^{2} + *y*^{2}/*b*^{2} = 1 one can adopt the idea for the orthoptic for the quadratic equation

Now, as in the case of a parabola, the quadratic equation has to be solved and the two solutions *m*_{1}, *m*_{2} must be inserted into the equation

Rearranging shows that the isoptics are parts of the degree-4 curve:

- Hyperbola:

The solution for the case of a hyperbola can be adopted from the ellipse case by replacing *b*^{2} with −*b*^{2} (as in the case of the orthoptics, see above).

To visualize the isoptics, see implicit curve.

Wikimedia Commons has media related to Isoptics.

*Special Plane Curves.*- Mathworld
- Jan Wassenaar's Curves
- "Isoptic curve" at MathCurve
- "Orthoptic curve" at MathCurve

- Lawrence, J. Dennis (1972).
*A catalog of special plane curves*. Dover Publications. pp. 58–59. ISBN 0-486-60288-5. - Odehnal, Boris (2010). "Equioptic Curves of Conic Sections" (PDF).
*Journal for Geometry and Graphics*.**14**(1): 29–43. - Schaal, Hermann (1977).
*Lineare Algebra und Analytische Geometrie*. Vol. III. Vieweg. p. 220. ISBN 3-528-03058-5. - Steiner, Jacob (1867).
*Vorlesungen über synthetische Geometrie*. Leipzig: B. G. Teubner. Part 2, p. 186. - Ternullo, Maurizio (2009). "Two new sets of ellipse related concyclic points".
*Journal of Geometry*.**94**(1–2): 159–173. doi:10.1007/s00022-009-0005-7. S2CID 120011519.